Deep Accurate Solver for the Geodesic Problem

TL;DR

A neural network-based local solver which implicitly approximates the structure of the continuous surface and supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods.

Abstract

A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon are at most second-order accurate with respect to the distances on the corresponding continuous surface. By order of accuracy we refer to the convergence rate as a function of the average distance between sampled points. Next, a higher-order accurate deep learning method for computing geodesic distances on surfaces is introduced. Traditionally, one considers two main components when computing distances on surfaces: a numerical solver that locally approximates the distance function, and an efficient causal ordering scheme by which surface points are updated. Classical minimal path methods often exploit a dynamic programming principle with quasi-linear computational complexity in the number of sampled points. The quality of the distance approximation is determined by the local solver that is revisited in this paper. To improve state of the art accuracy, we consider a neural network-based local solver which implicitly approximates the structure of the continuous surface. We supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods. The result is a third-order accurate solver with a bootstrapping-recipe for further improvement.

Figures (15)

• Figure 1: The proposed local solver pipeline described in Section \ref{['local_Solver_section']}. For each point in the Wavefront, the Visited points from from its local neighbourhood are chosen, and their coordinates and distance function are transformed into their canonical representation with respect to the target point. The processed input is fed into the neural network and the output is further processed to revert the centering and scaling transformations.
• Figure 2: The proposed network architecture described in Section \ref{['local_Solver_section']}. The input coordinates and distance function are in their canonical form, after translation rotation and scale.
• Figure 3: Bootstrapping by training. Distance values computed for a high $h^2$-resolution sampled mesh of a continuous surface with an $r$ accurate scheme yields $\mathcal{O} (h^{2r})$ accurate distances given at the mesh points. The mesh can then be sampled into a lower $h$-resolution mesh of the same continuous surface, while keeping the corresponding $\mathcal{O} (h^{2r})$ accurate distances at the vertices. See text for an elaborated discussion regarding data augmentation at high resolution and training more accurate update procedures at the low resolution.
• Figure 4: Order of accuracy: Right: Plots showing how the edge size effects the error. The accuracy of each scheme is defined by the corresponding slope. We used mesh approximations of a unit sphere with different edge size. Left: Errors presented on high resolution spheres for 2nd order methods and our presented method (when brighter colour indicates higher error).
• Figure 5: Polynomial surfaces: Errors presented for the polyhedral scheme and the proposed method. Local errors, represented as colors on the surface, were computed relative to exact polyhedral distances computed at a high-resolution sampled mesh of the continuous surface, as described in Subsection \ref{['generate_gt_section']}.

Theorems & Definitions (1)

• theorem 1